Symmetry

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19 pages, 354 KiB

Open AccessArticle

On the Analytic Continuation of Appell’s Hypergeometric Function F₂ to Some Symmetric Domains in the Space

C^{2}

by Roman Dmytryshyn

Symmetry 2024, 16(11), 1480; https://doi.org/10.3390/sym16111480 - 6 Nov 2024

Viewed by 752

Abstract

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of Appell’s hypergeometric function

F_{2}

with real and complex parameters, using [...] Read more.

The paper considers the problem of representation and extension of Appell’s hypergeometric functions by a special family of functions—branched continued fractions. Here, we establish new symmetric domains of the analytical continuation of Appell’s hypergeometric function

F_{2}

with real and complex parameters, using their branched continued fraction expansions whose elements are polynomials in the space

C^{2}

. To do this, we used a technique that extends the domain of convergence of the branched continued fraction, which is already known for a small domain, to a larger domain, as well as the PC method to prove that it is also the domain of analytical continuation. A few examples are provided at the end to illustrate this. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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17 pages, 16216 KiB

Open AccessArticle

Non-Standard Finite Difference and Vieta-Lucas Orthogonal Polynomials for the Multi-Space Fractional-Order Coupled Korteweg-de Vries Equation

by Khaled M. Saad and Rekha Srivastava

Symmetry 2024, 16(2), 242; https://doi.org/10.3390/sym16020242 - 16 Feb 2024

Cited by 1 | Viewed by 1094

Abstract

This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For [...] Read more.

This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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12 pages, 392 KiB

Open AccessArticle

Ulam–Hyers Stability of Linear Differential Equation with General Transform

by Sandra Pinelas, Arunachalam Selvam and Sriramulu Sabarinathan

Symmetry 2023, 15(11), 2023; https://doi.org/10.3390/sym15112023 - 5 Nov 2023

Cited by 7 | Viewed by 1296

Abstract

The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to [...] Read more.

The main aim of this study is to implement the general integral transform technique to determine Ulam-type stability and Ulam–Hyers–Mittag–Leffer stability. We are given suitable examples to validate and support the theoretical results. As an application, the general integral transform is used to find Ulam stability of differential equations arising in Thevenin equivalent electrical circuit system. The results are graphically represented, which provides a clear and thorough explanation of the suggested method. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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13 pages, 712 KiB

Open AccessArticle

A Note on the Lambert W Function: Bernstein and Stieltjes Properties for a Creep Model in Linear Viscoelasticity

by Francesco Mainardi, Enrico Masina and Juan Luis González-Santander

Symmetry 2023, 15(9), 1654; https://doi.org/10.3390/sym15091654 - 26 Aug 2023

Cited by 4 | Viewed by 1270

Abstract

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch,

W_{0} (t)

[...] Read more.

The purpose of this note is to propose an application of the Lambert W function in linear viscoelasticity based on the Bernstein and Stieltjes properties of this function. In particular, we recognize the role of its main branch,

W_{0} (t)

, in a peculiar model of creep with two spectral functions in frequency that completely characterize the creep model. In order to calculate these spectral functions, it turns out that the conjugate symmetry property of the Lambert W function along its branch cut on the negative real axis is essential. We supplement our analysis by computing the corresponding relaxation function and providing the plots of all computed functions. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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12 pages, 364 KiB

Open AccessArticle

Monomiality and a New Family of Hermite Polynomials

by Giuseppe Dattoli and Silvia Licciardi

Symmetry 2023, 15(6), 1254; https://doi.org/10.3390/sym15061254 - 13 Jun 2023

Cited by 10 | Viewed by 1264

Abstract

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities [...] Read more.

The monomiality principle is based on an abstract definition of the concept of derivative and multiplicative operators. This allows to treat different families of special polynomials as ordinary monomials. The procedure underlines a generalization of the Heisenberg–Weyl group, along with the relevant technicalities and symmetry properties. In this article, we go deeply into the formulation and meaning of the monomiality principle and employ it to study the properties of a set of polynomials, which, asymptotically, reduce to the ordinary two-variable Kampè dè Fèrièt family. We derive the relevant differential equations and discuss the associated orthogonality properties, along with the relevant generalized forms. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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13 pages, 287 KiB

Open AccessArticle

Poly-Cauchy Numbers with Higher Level

by Takao Komatsu and Víctor F. Sirvent

Symmetry 2023, 15(2), 354; https://doi.org/10.3390/sym15020354 - 28 Jan 2023

Cited by 1 | Viewed by 977

Abstract

In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher [...] Read more.

In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher levels are yielded from the inverse relationship with an s-step function of the exponential function. We show such a function with recurrence relations and give the expressions of poly-Cauchy numbers with higher levels. Poly-Cauchy numbers with higher levels can be also expressed in terms of iterated integrals and a combinatorial summation. Poly-Cauchy numbers with higher levels for negative indices have a double summation formula. In addition, Cauchy numbers with higher levels can be also expressed in terms of determinants. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

15 pages, 8858 KiB

Open AccessFeature PaperArticle

Bell’s Polynomials and Laplace Transform of Higher Order Nested Functions

by Diego Caratelli and Paolo Emilio Ricci

Symmetry 2022, 14(10), 2139; https://doi.org/10.3390/sym14102139 - 13 Oct 2022

Cited by 1 | Viewed by 1397

Abstract

Using Bell’s polynomials it is possible to approximate the Laplace Transform of composite functions. The same methodology can be adopted for the evaluation of the Laplace Transform of higher-order nested functions. In this case, a suitable extension of Bell’s polynomials, as previously introduced [...] Read more.

Using Bell’s polynomials it is possible to approximate the Laplace Transform of composite functions. The same methodology can be adopted for the evaluation of the Laplace Transform of higher-order nested functions. In this case, a suitable extension of Bell’s polynomials, as previously introduced in the scientific literature, is used, namely higher order Bell’s polynomials used in the representation of the derivatives of multiple nested functions. Some worked examples are shown, and some of the polynomials used are reported in the Appendices. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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18 pages, 470 KiB

Open AccessArticle

Implementation of Two-Mode Gaussian States Whose Covariance Matrix Has the Standard Form

by Gianfranco Cariolaro and Roberto Corvaja

Symmetry 2022, 14(7), 1485; https://doi.org/10.3390/sym14071485 - 20 Jul 2022

Cited by 3 | Viewed by 2583

Abstract

This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. In the two-mode states, the (ordinary) CM is a real symmetric matrix of order 4; therefore, it depends on 10 real [...] Read more.

This paper deals with the covariance matrix (CM) of two-mode Gaussian states, which, together with the mean vector, fully describes these states. In the two-mode states, the (ordinary) CM is a real symmetric matrix of order 4; therefore, it depends on 10 real variables. However, there is a very efficient representation of the CM called the standard form (SF) that reduces the degrees of freedom to four real variables, while preserving all the relevant information on the state. The SF can be easily evaluated using a set of symplectic invariants. The paper starts from the SF, introducing an architecture that implements with primitive components the given two-mode Gaussian state having the CM with the SF. The architecture consists of a beam splitter, followed by the parallel set of two single–mode real squeezers, followed by another beam splitter. The advantage of this architecture is that it gives a precise non-redundant physical meaning of the generation of the Gaussian state. Essentially, all the relevant information is contained in this simple architecture. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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16 pages, 407 KiB

Open AccessArticle

Gaussian States: Evaluation of the Covariance Matrix from the Implementation with Primitive Component

by Gianfranco Cariolaro, Roberto Corvaja and Filippo Miatto

Symmetry 2022, 14(7), 1286; https://doi.org/10.3390/sym14071286 - 21 Jun 2022

Cited by 1 | Viewed by 1930

Abstract

Quantum Gaussian states play a fundamental role in quantum communications and in quantum information. This paper deals with the implementation of multimode, and particularly of two-mode Gaussian unitaries and Gaussian states with primitive components (phase shifters, single-mode real squeezers, displacements, and beam splitters). [...] Read more.

Quantum Gaussian states play a fundamental role in quantum communications and in quantum information. This paper deals with the implementation of multimode, and particularly of two-mode Gaussian unitaries and Gaussian states with primitive components (phase shifters, single-mode real squeezers, displacements, and beam splitters). The architecture thus obtained allows one to obtain an insight into the physical meaning of each variable involved. Moreover, following the implementation architecture, it is possible to formulate an easy algebra (radical free) for the main operations and transformations of Gaussian states. Full article

(This article belongs to the Special Issue Theory and Applications of Special Functions II)

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